My favorite are the D&D box games-
Wrath of Ashardalon is a good’n.
Theres a series of dungeon tiles with various layouts on them shuffled into a deck, face down. As players “explore” the edges of the map- They place tiles down (each tile has an arrow on one corner, you connect it with the arrow facing the players piece)- Which can make for some nifty and random dungeon lay outs:
Tiles will indicate if you ran into enemies or not (Which come from a shuffled monster deck, and the game has miniatures included for all of them)
All while pulling from an “encounter” deck, which give you additional challenges to face. Sometimes its traps, persisting environmental effects, random treasures, or a good ol fashioned ambush.
Players control specific heroes with character cards explaining their abilities, stats (armor, hp, etc), and get to pick from a pool of abilities- Granting them “at will” abilities they can use in combat, utility powers which typically work as one-off buffs, and a powerful “daily” ability which is basically a single use, “but something bad ass will happen” ability. As you kill monsters you collect treasures to gear up, and advance your way through the dungeons.
In typical D&D fashion the goals/quests depend upon you. But if youre not feeling creative the game includes a book which has a handfull of pre-done quests to accomplish- and its pretty easy to concoct your own. Dont worry- its not as difficult as it sounds. Theres a few “unique” tile pieces, the general idea is to take one of these tiles, mix it with 3-4 other tiles, and then put them on the bottom of the tile deck (So itll happen ROUGHLY at a specific point, but you wont know exactly where), and have a preset idea when you encounter this tile- Maybe youre disarming a trap to collect a piece of treasure, maybe you come across an altar which summons the big baddie ashardalon himself (who has an amazing figure). Etc and etc.
If you like the idea of D&D, but find the rule book daunting- This is a GREAT way to get introduced to the “concept” of the game in a pretty stream-lined fashion. It really only takes about 30 minutes to an hour to familiarize yourself with the rule book, and quests- and youre good to go. Once you know the rules, you can easily get ANYONE else playing the game with about 2 minutes of “intro” (The game has a “step by step” card for each player, and it really is as easy as follow 1, 2, 3. Move your hero and complete an action if you wish, explore the dungeon if youre on the edge of a tile, go through the villain phase. tada).